Courses

Unlike their complex counterparts, equisingularity results for tame families of subsets are harder to come by in Real Tame Geometry. Nevertheless a promising, though not unexpected, approach is to walk in the world of integral geometry and get sufficient conditions in terms of continuity of certain functions built from curvature measures associated with the members of the family. Nicolas Dutertre will introduce us to his tools from Geometric measure theory (Lipschitz-Killing Measures) and how they naturally appear with good properties in equisingularity problems.

Lecture 1 (Tue, May 25, 09:00). On the topology of semi-algebraic sets: stratified critical points and an index theorem.

Notes of Lecture 1

Lecture 2 (Wed, May 26, 09:00). Lipschitz-Killing measures and Gauss-Bonnet theorems

Lecture 3 (Thu, May 27, 10:15). Lipschitz-Killing measures and Gauss-Bonnet theorems

Notes for Lecture 2 and Lecture 3

Lecture 4 (Fri, May 28, 10:15). Some results on real equisingularity.

Notes for Lecture 3 and Lecture 4

Regulous function (continuous extensions of rational functions) have appeared almost twenty years ago in the works of ​Kollár and Nowak. In the last six years they have been the focus of some serious interest. Goulwen Fichou will present us this category of functions and their specs as well as many current problems of Real Geometry where they naturally appear.

Lecture 1 (Mon, May 24, 10:15). What is a regulous function? Typical example on the real plane.

Notes for Lecture 1.

Lecture 2 (Tue, May 25, 10:15). Regulous functions: history, first properties, strange behavior on singular sets.

Notes for Lecture 2

Lecture 3 (Wed, May 26, 10:15). Regularity on a stratification and regularity after blowings-up.

Notes for Lecture 3

Lecture 4 (Fri, May 28, 09:00). Łojasiewicz property and real algebra: expansion of continuous function, Nullstellensatz, real algebraic versus regulous sets.

Notes for Lecture 4

We invited Olivier le Gal to present the fundamental notions of o-minimal Geometry, starting with the foundational example of the semi-algebraic structure. As a specialist in o-minimal geometry he will hint at how the four others courses are related.

Lecture 1 (Mon, May 24, 09:00). Semi-algebraic sets : definition and basic properties, Tarski, cell decomposition, dimension.

Main reference:

Michel Coste, An introduction to semi-algebraic geometry, Instituti editoriali e poligrafici inter- nazionali, 2000.

Notes for Lecture 1.

Black Board Lecture 1

Lecture 2 (Tue, May 25, 11:45). O-minimality : Definition of an o-minimal expansion of the field of real, finiteness properties, selection lemma, $C^k$.

Main references:

Michel Coste, An introduction to o-minimal geometry, Instituti editoriali e poligrafici inter- nazionali, 2000.

van den Dries, Lou Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, 248. Cambridge University Press, Cambridge

Notes for Lecture 2

Black Board Lecture 2

Lecture 3 (Wed, May 26, 11:45). Polynomial boundedness versus exponential: Chris Miller's dichotomy growth theorem, Hardy field of an o-minimal structure, asymptotic of definable functions, Łojasiewicz inequality in polynomially bounded structures.

Main references:

Chris Miller, Exponentiation is hard to avoid. Proc. Amer. Math. Soc. 122 (1994), no. 1, 257–259.

Chris Miller, Basics of o-minimality and Hardy fields. Lecture notes on o-minimal structures and real analytic geometry, 43–69, Fields Inst. Commun., 62, Springer, New York, 2012.

Notes for Lecture 3

Black Board Lecture 3

Lecture 4 (Thu, May 27, 09:00). Quasi-analytic classes and o-minimality: In this session we deal with the link between quasi-analytic classes and o-minimality. We review some examples of o-minimal structures generated by quasi-analytic classes.

Main reference:

Rolin J.-P., Speissegger P., Wilkie, A. J., Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), no. 4, 751–777.

Notes for Lecture 4

Black Board Lecture 4

Ultra-differentiable Analysis concerns sub-algebras of smooth functions with constrained growth of the Taylor series coefficients. Besides its importance in the analysis of partial differential equations the development of this theory was influenced by the classical Whitney extension problem and the composition problem. Both problems followed a path that often meets sub-analytic geometry, and later o-minimal geometry. Of special interest among the ultra-differentiable classes are the quasi-analytic classes which possess a (quasi-)analytic continuation property similar to the real analytic class. This property make these classes interesting from an analytic viewpoint and also suitable for questions of tame geometry. In relation with asymptotic expansions of solutions of analytic ODEs they also naturally appear when dealing with non-oscillation problems of the solutions. Armin Rainer will introduce us to ultra-differentiable classes, in particular to quasi-analytic ones.

The mini-course is intended as an introduction to ultradifferential analysis with special emphasis on ultra-differentiable extension theorems. The development of differential analysis in the last century was decisively influenced by Whitney’s work on the extension of differentiable functions from closed sets. We shall be interested in quantitative versions of Whitney’s extension theorem. The quantitative aspect is implemented by uniform growth properties of the multisequence of partial derivatives which measure the deviation from the Cauchy estimates and hence from analyticity. These growth conditions determine ultradifferentiable function classes which form a scale of regularity classes between the real analytic and the smooth class.

Lecture 1 (Mon, May 24, 13:00). We will recall Whitney’s classical extension theorem and formulate the quantitative problem. This will lead us to Denjoy–Carleman classes which are the ultradifferentiable classes of main interest in this series of lectures. After discussing inclusion and stability properties we shall begin with the study of the Borel map (i.e. infinite Taylor expansion) on Denjoy–Carleman classes. We will investigate when the Borel map is injective and, in the course of this, prove the Denjoy–Carleman theorem which discriminates between quasianalytic and non-quasianalytic classes.

Slides of Lecture 1.

Lecture 2 (Tue, May 25, 13:00). Next we will study the image of the Borel map on Denjoy–Carleman classes which naturally sits in the sequence space defined by the characteristic bounds. The quasianalytic and the non-quasianalytic case are fundamentally different. While in the non-quasianalytic case we will give necessary and sufficient conditions for the Borel map being onto the sequence space, we shall see that on quasianalytic classes, strictly containing the real analytic class, the Borel map is never onto. This will be complemented by some results on the description of the Borel image and a discussion of further intricacies of the quasianalytic setting.

Slides Of lectures 2

Lecture 3 (Thu, May 27, 11:45). The third lecture is devoted to the Whitney extension problem for arbitrary closed sets. The key ingredient is the existence of optimal cut-off functions which yield, in combination with a family of Whitney cubes, a suitable partition of unity on the complement of the closed set. We shall see that this partition of unity allows us to glue the local extensions for the singleton (Borel map) to a global extension. We will also discuss the existence of continuous linear extension operators which is intimately related to interesting topological invariants of the involved function spaces.

Notes for Lecture 3

Lecture 4 (Fri, May 28, 11:45). An ultradifferentiable Whitney approximation theorem will enable us to conclude that the extension can always be chosen real analytic off the given closed set. Furthermore, we will discuss the extension problem with a controlled loss of regularity. This requires a different approach which is closely related to the characterization of ultradifferentiable classes by almost analytic extensions. We will address several applications such as Lojasiewicz’s theorem on regularly situated sets or Whitney’s spectral theorem in the ultradifferentiable framework. Finally, we shall take a glimpse on other ultradifferentiable classes, in particular, Braun–Meise–Taylor classes.

Notes for Lecture 4

A very interesting setting where Real Analytic Geometry interacts with Model Theory concerns the asymptotic behaviour of trajectories of real analytic vector fields at singular points, either as a single object or as pencils of such objects. Little is known is general and, yet, there are many partial results and examples about this appealing topic. Fernando Sanz will tell us more about this, and if times allows will be able to speak of some of his latest results and share a few hopes for still open problems.

Preliminary course notes (in spanish).

Lecture 1 (Mon, May 24, 11:45). Notes for Lecture 1.

Lecture 2 (Wed, May 26, 13:00). Notes for Lecture 2

Lecture 3 (Thu, May 27, 13:00). Notes for Lecture 3

Lecture 4 (Fri, May 28, 13:00). Notes For Lecture 4